{
  "id": "1505.03089",
  "version": "v1",
  "published": "2015-05-12T17:17:19.000Z",
  "updated": "2015-05-12T17:17:19.000Z",
  "title": "Quaternionic R transform and non-hermitian random matrices",
  "authors": [
    "Zdzislaw Burda",
    "Artur Swiech"
  ],
  "comment": "21 pages, 6 figures",
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "math.MP"
  ],
  "abstract": "Using the Cayley-Dickson construction we generalize the free probability calculus to non-hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (non-crossing) cumulants of the given random matrix $X$ in the limit of infinite matrix size $N\\rightarrow \\infty$. The cumulants are defined as connected averages of all distinct powers of $X$ and its hermitian conjugate $X^\\dagger$ of the type: $\\langle\\langle \\frac{1}{N} \\mbox{Tr} X^{a} X^{\\dagger b} X^c \\ldots \\rangle\\rangle$ for $N\\rightarrow \\infty$. We show that the R transform for gaussian elliptic laws is given by a simple linear quaternionic map $\\mathcal{R}(z+wj) = x + \\sigma^2 \\left(\\mu e^{2i\\phi} z + w j\\right)$ where $(z,w)$ is the Cayley-Dickson pair of complex numbers forming a quaternion $q=(z,w)\\equiv z+ wj$. This map has five real parameters $\\Re e x$, $\\Im m x$, $\\phi$, $\\sigma$ and $\\mu$. We use the R transform to formulate the addition and multiplication laws for non-hermitian random matrices. For illustration we apply these laws to calculate the limiting eigenvalue densities of several products of gaussian random matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-12T17:17:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random matrix",
      "non-hermitian random matrices",
      "simple linear quaternionic map",
      "gaussian elliptic laws",
      "free probability calculus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}